The Sagnac experiment was sufficient proof of the inequality
(and indirect evidence for the classic law of addition of velocities). Recall
that four mirrors (more exactly three mirrors and one plate - see
Fig. 3.4) were installed along the periphery of a disc rotating at
angular rate .

Figure 3.4:
The Sagnac experiment.

A light beam was divided (by the plate ) into two beams, and one beam
traveled counterclockwise (in the direction of rotation) while the other
traveled clockwise. An interference was observed at meeting of these
beams. The fringe shift (as a result of the difference in times of
propagation of light beams) had magnitude: .
It is obvious that the non-inertial character of the system rotating at
is of no concern: nobody saw a curved light beam in vacuum;
light travels between two reflections rectilinearly. Nevertheless,
we consider the following mental experiment: Imagine that the disc radius tends
to infinity , but the value remains constant.
Then we have . Therefore, the value of the acceleration
tends to zero. Let us choose a radius such that the
acceleration is much less than any pre-specified value (the existing
experimental accuracy, for example). Nobody can distinguish
this "near-inertial" system from a true inertial system.
If the number of equidistant mirrors is also increased (),
then the straight line (of light beams) between mirrors approaches
the disc circle. As a result the
fringe shift can be expressed as ,
where is a constant for a given and is the circumference.
Because of the obvious symmetry of the experiment, the effect is
additive in , and its value can be related to the unit length. A
"cumulative" effect of acceleration can be made less than any pre-specified
value for a given straightline region. Thus, we have for the magnitude of
the fringe shift: (some variations in produce
appropriate variations in , since is a finite value).
Therefore, the time of signal propagation linearly depends on the velocity of
the motion of the system, that is, .